Browsing by Author "Ranirina, Dinna"

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    Refinable vector splines and multi-wavelets with shortest matrix filters
    (Stellenbosch : Stellenbosch University, 2018-03) Ranirina, Dinna; De Villiers, Johan; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT : A widely used class of basis functions in signal analysis is obtained from the dilation and integer shifts of a given (compactly supported) wavelet ψ : R → R, by means of which a (scalar) signal can be decomposed into its low frequency and high frequency components. Whereas initially much attention was devoted to orthogonal wavelet decomposition techniques (see for example [1] and [2]), the recent book [3] introduced a more general approach to wavelet construction in which orthogonality is not a requirement and which yielded signi cant advantages in some application areas. An interesting extension is to consider instead, with the view to the decomposition of a vector-valued signal, as presented for the orthogonal case in, for example, [4], a multi-wavelet Ψ : R → R ν . The main focus of this study is to extend the methods in [3], in order to characterize, by means of matrix Laurent polynomial identity systems, a class of multi-wavelets based on general (not necessarily orthogonal) space decomposition. As main building blocks are used re nable vector functions , together with their corresponding matrix re nement sequences. Three di erent classes of re nable vector splines are analysed, with particular focus also on their integer-shift linear independence and stability properties, before explicitly constructing their corresponding spline multi-wavelets. The low-pass and high-pass decomposition matrix lter sequences thus obtained are the shortest possible for the given re nable vector spline, and the spline multi-wavelet is of minimal support for these optimal matrix lters. Moreover, our approach yields explicit formulations for the re nable vector splines, as well as for their corresponding spline multi-wavelets and matrix lter sequences. Computationally e cient algorithms are developed, and examples are calculated, with accompanying illustrating graphs.